A GENERALIZATION OF THE JOINT SPECTRAL RADIUS: THE GEOMETRICAL APPROACH

The generalized joint spectral radius with exponent p is investigated. 1. Basic Notions Definition 1. The joint spectral radius of a given set {A1, A2, . . . , Ak} ⊂ L(Rd) of linear operators is ρ(A1, A2, . . . , Ak) = lim m→∞max σ ‖Aσ(1) · · ·Aσ(m)‖ 1/m , where σ runs the set of all the substitutions σ: {1, . . . ,m} → {1, . . . , k}. The notion of joint spectral radius have appeared birthly by G.C. Rota and G. Strang in 1960 [1]. In 1988–90 its close relation to the regularity exponent of Daubeshies wavelets in the space C[a, b] was established by Collela, Heil, Daubeshies, Lagarias etc. [2], [3], [4], [5]. The wavelets of Daubeshies is a system of functions {ψij}i,j∈Z 1) ψij = 2−j/2ψ00(2jx− i), 2) {ψij} – ortonormal basis in L2(R), 3) suppψ00 ⊂ [0, N ], N ∈ N. The function ψ00 can be constructed by a scaling function φ by formula ψ00(x) = N ∑ k=0 (−1)CN−kφ(2x − k). The scaling function φ is a solution of dilation equation or two-scale difference equation: